Legal matters always depend on the jurisdiction. Here in Finland, for example, everyone has the right to privacy and "secrecy of confidential message". The law has a couple exceptions, but things like age or who pays for devices are not among them. A parent may only read a specific message addressed to their child if there is justifiable reason to believe that reading that specific message is necessary for the child's safety.
Of course it's pretty rare for children to actually press charges against their parent for that, but it has happened. Using computer software "designed or modified to" violate the law is also specifically mentioned to make it an aggravated offense, with maximum penalty of 3 years in prison.
